Towards a classification of connected components of the strata of $k$-differentials

TL;DR

This paper advances the classification of connected components in the strata of $k$-differentials on Riemann surfaces, extending known results to general $k$ and providing new techniques and applications, including a complete classification for quadratic differentials with arbitrary poles.

Contribution

It develops new methods to classify connected components of $k$-differential strata for all $k$, generalizing previous results and introducing techniques like multi-scale $k$-differentials.

Findings

Complete classification of connected components for quadratic differentials with arbitrary poles.

Distinction of certain strata components via hyperelliptic structure and spin parity for higher $k$.

Approach to determine parities of $k$-differentials in genus zero and one, leading to a number theory conjecture.

Abstract

A $k$-differential on a Riemann surface is a section of the $k$-th power of the canonical bundle. Loci of $k$-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification for the moduli space of $k$-differentials. The classification of connected components of the strata of $k$-differentials was known for holomorphic differentials, meromorphic differentials and quadratic differentials with at worst simple poles by Kontsevich--Zorich, Boissy and Lanneau, respectively. Built on their work we develop new techniques to study connected components of the strata of $k$-differentials for general $k$. As an application, we give a complete classification of connected components of the strata of quadratic differentials with arbitrary poles. Moreover, we distinguish certain components of the strata of $k$-differentials by generalizing the hyperelliptic structure and spin parity for higher $k$. We also describe an approach to determine explicitly parities of $k$-differentials in genus zero and one, which inspires an amusing conjecture in number theory. A key viewpoint we use is the notion of multi-scale $k$-differentials introduced by Bainbridge--Chen--Gendron--Grushevsky--M\"oller for $k = 1$ and extended by Costantini--M\"oller--Zachhuber for all $k$.